In Strogatz' 2nd column, he discusses how thinking of numbers (positive integers, anyway) as groups of rocks can make what seems boring more approachable and interesting.
For example, you might notice that you can make different kinds of shapes with your rocks, like squares and triangles.
(The 4 means that it's a square with 4 rocks on a side. The 3 means it's a triangle with 3 rocks on a side).
But only certain numbers of rocks can make certain shapes.
It's easy to get curious. The triangular numbers (in red) go 3, 6, 10, 15. What is it about these numbers that lets you make triangles out of them? And what about 4, 9, 16 lets you make squares? If you think about it, the number 1 is sort of like the smallest square AND the smallest triangle. Are there any other numbers that you could make both a square and a triangle out of?
Using rock piles, you can fiddle with these questions and discover lots of cool number patterns visually. For example, you might notice that you can build a triangular number by adding increasing integers starting at one.
Think about that. All triangles are increasing sums.
You may also notice that you can get a square number by adding two triangular numbers.
In other words...
Triangle_5 + Triangle_4 = Square_5
And each triangle is just an increasing sum. SO...
Triangle_5 + Triangle_4 = Square_5
(1 + 2 + 3 + 4 + 5) + (1 + 2 + 3 + 4) = 5*5
And it's a pretty sure bet that the pattern will continue...
(1 + 2 + 3 + 4 + 5 + 6) + (1 + 2 + 3 + 4 + 5) = 6*6
This kind of thing can be useful as well. There's a legend about the mathematician and physicist Carl Gauss. When he was in 2nd grade, as the story goes, he was acting up and his teacher told him to find the sum of all the numbers from 1 to 100. Instead of adding them, as was expected, he took advantage of a pattern and solved the problem in under a minute.
Maybe you can see it too if we look at what that sum would look like as a group of rocks.
One pattern you might notice is that the sum looks like half of a square.
As you can see, it's actually a little more than half of 10^2, because there's all those half-circles that cross the half-way line. There are 10 half circles there, so that's really 5 whole circles.
That means that....
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10*10 + 5
That pattern makes it easy to do the same thing for the sum of 1 to 100. By the same reasoning, the answer should be 100*100 + 50.
Or you might see a different kind of pattern. Imagine taking the lower triangle here...
And moving it to create a rectangle.
This shows that the same sum is really 11 * 5!
Pretty clever! So sometimes playing around with number patterns turns out to be useful. In fact, that could be the subtitle for The Story Of Math. Time and time again, mathematicians who enjoy playing with patterns discover something interesting...and then 100 or 200 years later, the pattern turns out to be incredibly useful for something.
This visual way of thinking about numbers can also help you avoid a mistake that almost every high school student makes at one time or another.
If you had to make a guess about what (6 + 5)^2 is, what would it be?
A pretty reasonable guess might be 6^2 + 5^2. But is it true? Don't reach for your calculator!
It's very easy to check if you just imagine what shapes the numbers make.
Nope! Clearly the 6^2 and 5^2 only make up a part of (6 + 5)^2
How much more is needed to fill in the whole square? It looks like the two faint rectangles are each 6*5. So, we have that....
Big Square = Blue Square + Red Square + Two Rectangles
(6 + 5)^2 = 6^2 + 5^2 + 2*(6*5)
When I was in school, the way you figured out (a + b)^2 was by using an acronym: FOIL. It stands for First-Outer-Inner-Last, which is supposed to describe how you mechanically multiply the numbers to get the answer. It is also notoriously difficult for some students.
Area models not only make it easier, but also make it obvious WHY they work.
SO, for me, the key idea here is: Having a physical or visual representation for math concepts makes them easier to understand, get curious about, and play with. These patterns are really numerical patterns. The visual model just helped us see them more clearly.
There's nothing really magic about rock groups. If you decide to leave groups of rocks behind and just think about the area of a square, you can get even more surprising results.
In this picture, an infinite number of squares and rectangles representing successively smaller fractions can all fit inside a square with an area of 1. It's kind of surprising that an infinite sum of numbers could have a finite total, but there it is!
Using visual models and metaphors is not without its dangers, however. We'll look at a few of those next time...
